Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-17T13:43:47.730Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral theory and Fuchsian groups

Published online by Cambridge University Press:  24 October 2008

S. J. Patterson
S. J. Patterson
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we return to consider more fully some questions which were touched on in (6). This was the first of a series of papers dealing with the spectral theory of the Laplace–Beltrami operator on a Riemann surface. In the later parts of that series we dealt fully with the case when the fundamental group is finitely generated. In this paper we return to the study of more general questions. In many ways this work continues that of W. Roelcke (7). We shall develop in section 2 the spectral theory for the Laplace operator Δ operating on functions on G\D where D is the unit disc and G is an arbitrary Fuchsian group. The main object is to find generalized eigenfunction expansions which are valid in the sense of absolute convergence rather than L2 convergence. The most convenient form of spectral theory for our purposes appears to be a version due to Gårding for Carleman operators which is presented in ((4), Ch. xvii, §§ 8,9). As a first application we use the results in section 3 to solve a problem of W. K. Hayman. Then we turn to Fuchsian groups G with exponent of convergence δ(G) < ½. Using the results of (6) we first obtain sufficient information on the action of G on the principal circle of G (section 4) and finally determine the spectral resolution in section 5.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 1 , January 1977 , pp. 59 - 75
Copyright
Copyright © Cambridge Philosophical Society 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Elstrodt, J.Über das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene bei Fuchsschen Gruppen zweiter Art. Dissertation, München, 1969.Google Scholar
(2)Elstrodt, J.Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, Math. Ann. 203 (1973), 295–330; II, Math. Z. 132 (1973), 99–134; III, Math. Ann. 208 (1974), 99–132.CrossRefGoogle Scholar
(3)Kubota, T.Elementary theory of Eisenstein series (Kodansha Ltd, Tokyo, 1973).Google Scholar
(4)Maurin, K.Methods of Hilbert spaces (Warsaw, 1972).Google Scholar
(5)Patterson, S. J.A lattice-point problem in hyperbolic space. Mathematika 22 (1975), 81–88.CrossRefGoogle Scholar
(6)Patterson, S. J.The Laplacian operator on a Riemann surface. Comp. Math. 31 (1975), 83–107.Google Scholar
(7)Roelcke, W.Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, Math. Ann. 167 (1966), 292–337; II. Math. Ann. 168 (1967), 261–324.CrossRefGoogle Scholar
(8)Selberg, A.Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20 (1956), 47–87.Google Scholar
(9)Siegel, C. L.Topics in complex function theory, vol. 2 (Interscience, 1971).Google Scholar
(10)Tsuji, M.Potential theory in modern function theory (Maruzen Ltd, Tokyo, 1959).Google Scholar